Computer controlled work tool apparatus and method

ABSTRACT

A polishing machine is described in which a surface treatment tool is moved across the surface of a workpiece in accordance with a predefined tool-path, in order to carry out the desired treatment process. The tool-path is non-periodic and preferably pseudo-random. Various techniques are described for generating data representing the tool-path to be followed. A technique is also described for determining optimum control parameters used to control the polishing machine for a given tool-path. The surface treatment may be a shaping technique in which material is removed from the surface, or a technique for adding material to the surface of the workpiece, or a technique for modifying the surface or a region under the surface of the workpiece.

This Application is a Divisional Application claiming the benefit of,and priority to, U.S. patent application Ser. No. 12/674,952 titled“COMPUTER CONTROLLED WORK TOOL APPARATUS AND METHOD”, which is a 371National Stage Application of International Application No.PCT/GB08/02890 titled “COMPUTER CONTROLLED WORK TOOL APPARATUS ANDMETHOD” filed on Aug. 26, 2008, which claims priority to United KingdomPatent Application No. GB 0716597.0 titled “COMPUTER CONTROLLED WORKTOOL APPARATUS AND METHOD” filed on Aug. 24, 2007, each of which areincorporated by reference in their entirety.

The present invention generally relates to methods and apparatus forperforming surface treatments, and is principally concerned withcomputer controlled work tools and methods for working on or under thesurface of a workpiece. The invention has particular, although notexclusive, relevance to polishing machines and the like for abrading orpolishing the surface of a workpiece. The workpiece surface may be acurved or flat surface of an optical component such as a lens or amirror, or a screen component for a display device. The inventionfurther has utility in the levelling and polishing of silicon wafersprior to the manufacture of semiconductor devices from the wafers byconventional production techniques. Aspects of the invention haveutility in other fields, to provide techniques of addressing or treatingall parts of an extended surface area using a localised treatment devicewhile avoiding the creation of treatment artefacts over the extendedsurface.

The abrading or polishing of a surface of a workpiece is a techniquewhich has applications in many different fields including the productionof semi-conductor devices and optical components such as optical lenses.The requirement is to provide a surface which has a particular surfacecontour and a particular surface finish (e.g. smoothness). In the fieldof optical polishing there are two different techniques. One techniqueuses a tool for polishing which has a size similar to the size of theworkpiece. The limitation of this technique is that the tool is designedfor a specific workpiece and thus cannot be used universally.

In the second technique, the tool is substantially smaller than theworkpiece and is moved across the workpiece in accordance with apredefined tool-path, in order to carry out the abrading or polishingprocess. Such a technique is described in U.S. Pat. No. 4,128,968 and inWO 00/32353. In these prior art systems, the tool is moved across theworkpiece in accordance with a raster or spiral tool-path. Depending onthe difference between the actual surface profile of the workpiece andthe desired surface profile, the characteristics of the toolhead arevaried (e.g. pressure applied, speed of travel, dwell time etc.) inorder to obtain the desired surface profile.

The inventors, however, have identified that such a regular or periodictool-path creates rows of grooves and/or ridges resembling a diffractiongrating in the surface of the workpiece.

According to one aspect, the present invention provides a method andapparatus for performing a surface treatment on an area, by moving asurface treatment device along an irregular path which visits all pointsof the area. Preferably the path is a random or pseudo-random path. Morepreferably, the path visits all points of the area without visiting anypoint twice. Most preferably, the distance between neighbouringstretches of the path is arranged to be within a predetermined range ofdistances. In one embodiment, the path is arranged so that the distancebetween neighbouring stretches of the path is substantially the same forall or substantially all points on the path. In another embodiment, thepath is arranged so that the distance between neighbouring stretches ofthe path is reduced at some points along the length of the path ascompared to other points. Particularly, the surface treatment may becontrolled accurately when the distance between neighbouring stretchesof the path is reduced, and the tool footprint is correspondinglyreduced. Such accurate control is particularly beneficial in areasadjacent the edge of the workpiece.

The degree of treatment performed on an area of the workpiece may bevaried by varying the speed of movement of the tool along the tool pathin order to vary the dwell time of the tool at different points alongthe tool path. In an area where more treatment is required, the speed ofthe tool along the tool path is reduced at points on the tool path whichare within the area requiring more treatment. Alternatively, the toolmay be moved at a constant speed along the tool path and the “packingdensity” of the tool path (i.e. the spacing between neighbouringstretches of the tool path) may be increased in the area where moretreatment is required. In a further alternative, control of the degreeof treatment to which the workpiece is subjected in a particular areamay be affected using a combination of these two techniques, i.e. byvarying the speed of movement of the tool along the tool path, and byvarying the “packing density” of the tool path over the surface area tobe treated. In a yet further alternative, a variation of packing densitymay be combined with a variation in influence function. For example, acloser packing-density may be combined with a narrower influencefunction (smaller tool footprint), where it is required to minimiseeffects of edge-roll.

In one embodiment, measurements are taken at a plurality of points onthe surface, and an error map is generated to show a difference betweenthe actual measured surface and the required or target surface at eachpoint on the surface to be treated. The influence function of the toolis then varied as the tool travels across the surface, so that at eachpoint on the surface the error between the actual surface and the targetsurface is reduced to below a threshold value.

In an alternative embodiment, the tool is moved over the workpiecesurface following the tool path, keeping the influence function of thetool substantially constant so that each part of the surface receivessubstantially the same treatment. When a variation in the amount oftreatment delivered at different areas of the workpiece surface isrequired, this may be achieved using a tool of constant influencefunction by varying the “packing density” of the tool path, to reducethe spacing between neighbouring stretches of the tool path in areaswhere more surface treatment is required.

A second aspect of the invention provides a computer-controlled tool forworking on or under the surface of a workpiece by moving a surfacetreatment tool over the surface along a predefined tool-path which isnon-periodic and which visits all points of the area. In one embodiment,the tool-path is substantially random (pseudo-random) and generatedusing, for example, an artificial ant or a maze algorithm. In apreferred embodiment, the tool-path does not cross itself whichfacilitates the control of controllable characteristics of the tool asit is passed over substantially the entire surface of the workpiecebeing worked.

This aspect of the invention can be applied to a variety of differenttypes of computer-controlled work tools which are used for finishing thesurface of a workpiece. For example, it can be applied to systems wherethe tool removes material from the surface of the workpiece (such as bygrinding or polishing), or to tools which add material to the surface ofthe workpiece (such as by plating or coating). This aspect of theinvention also applies to tools which modify the surface (or a regionunder the surface) of the workpiece chemically or physically (e.g.hardening, ion-implantation, radioactive bombardment etc.) or to a toolwhich can be used to perform any combination of: adding material to,removing material from, or modifying the chemical or physical structureof the workpiece.

The invention also relates to an apparatus and method for generating anirregular path which addresses all parts of an extended surface.Preferably the path is a random or pseudo-random path. More preferably,the path visits all points of the area without visiting any point twice.The distance between neighbouring stretches of the path may be arrangedto be within a predetermined range of distances. In one embodiment, thepath is arranged so that the distance between neighbouring stretches ofthe path is substantially the same for all or substantially all pointson the path. In an alternative embodiment, however, the path is arrangedso that the distance between neighbouring stretches of the path isvaried at points along the path. In a particular embodiment, thedistance between neighbouring stretches of the path is reduced in areasadjacent the edge of the extended surface. The path may be used as atool-path for use in the machines referred to above, or may be used as alocus for sequentially addressing locations on a surface in asubstantially random manner.

These and various other aspects and features of the invention willbecome apparent from the following exemplary embodiments which aredescribed in detail with reference to the accompanying figures in which:

FIG. 1 is a perspective view of a polishing apparatus using a tool whichis moved over the surface of the workpiece;

FIG. 2a schematically illustrates one way in which the tool may bearranged to polish the surface of the workpiece;

FIG. 2b schematically illustrates an alternative arrangement of the toolfor polishing the surface of the workpiece;

FIG. 2c is a graph illustrating the removal profiles of the tool for theorientations shown in FIGS. 2a and 2 b;

FIG. 2d illustrates the removal profiles for the tool for differentoperating characteristics of the tool;

FIG. 3 is a schematic block diagram illustrating the main components ofthe computer-controlled tool used for polishing the surface of theworkpiece;

FIG. 4a schematically illustrates the surface of the workpiece to beworked and illustrating a non-periodic tool-path along which the tool isto be moved relative to the workpiece;

FIG. 4b illustrates in more detail part of the tool-path shown in FIG.4a and illustrating a mesh of points arrayed over a model of the surfaceof the workpiece, which is used in the calculations performed by thetool-path effect processor;

FIG. 4c illustrates in more detail some of the mesh points shown in FIG.4b and illustrating an area that is associated with a mesh point;

FIG. 5 is a flowchart illustrating the way in which a tool-path effectprocessor determines an influence matrix defining the influence of thetool on the workpiece for a given tool-path;

FIG. 6 schematically illustrates the form of a selected tool profile andillustrates some of the calculations carried out by the tool-path effectprocessor shown in FIG. 3;

FIG. 7 schematically illustrates the form of the influence matrixgenerated by the tool-path effect processor shown in FIG. 3;

FIG. 8 is a schematic block diagram illustrating in more detail the maincomponents of an optimiser which forms part of the system shown in FIG.3;

FIG. 9 illustrates the form of an actuator parameter matrix that is tobe optimised by the optimiser shown in FIG. 8;

FIG. 10 illustrates the form of a removal map that is calculated by theoptimiser using the influence matrix generated by the tool-path effectprocessor and a current actuator parameter matrix;

FIG. 11 illustrates the surface profile of the workpiece whenrepresented by two-dimensional parametric curves;

FIG. 12 illustrates a number of the control points for the workpieceillustrated in FIG. 11 and the areas associated with the control points;

FIG. 13 is a block diagram illustrating the main components of atool-path generator used to generate the tool-path data that defines thetool-path along which the tool is to be moved relative to the surface ofthe workpiece;

FIG. 14a schematically illustrates two candidate paths and a cuttingpoint at which the two candidate paths are cut into two segments;

FIG. 14b schematically illustrates two new candidate paths generated bysplicing together different segments from the two candidate paths shownin FIG. 14 a;

FIG. 15 illustrates the form of an exemplary stippling pattern; and

FIG. 16 illustrates the main components of an alternative tool-pathgenerator used to define tool-paths based on one or more stipplingpatterns obtained from a stippling pattern database.

FIRST EMBODIMENT

Overview

FIG. 1 is a perspective diagram of a polishing machine according to oneembodiment of the present invention. The polishing machine comprises atable 1 which is resistant to vibrations. On the table 1 there ismounted an X-slide mechanism 2 for movement in the x-direction. On theX-slide mechanism 2 there is mounted a Y-slide mechanism 3 for movementin the y-direction. On the Y-slide mechanism 3 there is mounted aturntable 4 for rotation in the direction indicated by arrow c. Theturntable 4 is mounted on the Y-slide mechanism 3 via a Z-movementmechanism (not shown) for movement of the turntable 4 in thez-direction. The turntable 4 has a support surface onto which aworkpiece 5 is mounted and held for polishing or abrading. Thus, thisarrangement provides for motion of the workpiece 5 in four axes—namelyx, y, z and c.

The polishing machine is also provided with a back member 6 on which ismounted a pivot arrangement for pivotally moving a polishing head 7. Thepolishing head 7 is arranged for axial rotation and includes a workingmember (or tool) 8 arranged at a lower axial end for polishing orabrading the workpiece 5. Thus axial rotation of the working member 8provides another axis for control, namely h.

The pivot mechanism mounted on the back member 6 comprises a first pivotmember 700 mounted in an arm for pivoting the head about a pivot pointin the working member 8 in a first plane. The first pivot mechanism 700is mounted on a second pivot mechanism 800 which provides for thepivoting of the head 7 about a pivot point in a plane perpendicular tothe plane of pivoting of the first pivot mechanism 700 in the arm. Thusthese two orthogonal pivoting mechanisms provide two further axes ofcontrol, namely a and b.

The back member 6 of the polishing machine also houses a computercontrol system 9 which includes a display 10 and control inputs 11. Thisallows a user to input controls to control the motion of the workpiece 5and of the tool 8 and to view displayed information regarding thepolishing or abrading process.

Each of the axes of motion x, y, z, c, h, a and b are driven byrespective drive actuators (not shown). Sensors (not shown) are alsoprovided for sensing the position of the actuators to provide positioninformation for use by the computer control system 9 to control thepolishing or abrading process. The computer control system 9 is alsoprovided with two further axes of control: namely the work done by thedriving mechanism in rotating the tool 8 held in the head 7 and thepressure applied within the tool 8. Thus, the computer control system 9operates an algorithm to control these nine axes (or parameters) inorder to abrade or polish the workpiece 5 mounted on the turntable 4, toachieve the desired surface profile and/or surface quality (such assmoothness). As will be apparent to those skilled in the art, thepolishing machine can be used to achieve any desired surface profileincluding a surface profile containing both concave and convex areas.

As will be described in more detail below, in this embodiment, thecomputer control system 9 is arranged to control the movement so thatthe tool 8 is arranged to follow a pseudo-random tool-path over thesurface of the workpiece 5 and controls the amount of polishing over thesurface by: (i) varying the time spent at each point along the tool-path(the dwell time), (ii) varying the transverse speed along the tool-path,and/or (iii) varying the rate of working of the tool 8. As a result ofthe use of a pseudo-random tool-path, polishing artefacts that aregenerated using a periodic spiral or raster tool-path can be avoided.

FIG. 2a schematically illustrates the surface of the workpiece 5 and thetool 8 used in this embodiment. As shown, in this embodiment the tool 8has a sheet of soft (deformable) polishing material 110 attached to itsend. To achieve the polishing, the polishing material 110 is broughtdown into contact with the surface of the workpiece 5 and the tool 8 isrotated about its axis 112. In the illustration of FIG. 2a , the tool 8is oriented so that its axis 112 lies normal to the surface of theworkpiece 5. Alternatively, as illustrated in FIG. 2b , the polishingmachine may be arranged to present the tool 8 at an angle (θ) to thesurface normal.

As a result of the rotation of the tool 8, the polishing material 110polishes (i.e. removes material from) the area of the workpiece 5 withwhich it is in contact. The amount of material removed within this areaof contact is not uniform and depends, among other things, on the angleθ. The amount of material removed also depends on the time that thepolishing material polishes the contact area. FIG. 2c illustrates, withthe solid line, 114 the removal profile (per microsecond) of the tool 8when the rotation axis 112 of the tool 8 is perpendicular to the surfaceof the workpiece 5. As shown, because there is no relative translationmovement between the tool and the workpiece at the centre of the area ofcontact, the removal rate at the centre is zero. However, by tilting thetoolhead so that it rotates at an angle θ to the surface normal, theresulting removal profile does not have this zero removal rate at thecentre, as illustrated by the dashed profile 116 shown in FIG. 2 c.

As those skilled in the art will appreciate, the removal profilesillustrated in FIG. 2c represent cross-sections of three-dimensionalremoval profiles defined, in part, by the three-dimensional shape of thetool 8. Additionally, in this embodiment, because a deformable sheet ofpolishing material 110 is used to perform the polishing, the removalprofile within the contact area is unlikely to be symmetric and willdepend on the surface geometry of the workpiece 5 within the contactarea.

In this embodiment, the polishing machine can vary the internal pressurewithin the tool 8 in order to control the pressure with which thepolishing machine applies the polishing material 110 to the surface ofthe workpiece 5. The removal profile for the tool will also depend uponthis pressure. FIG. 2d schematically illustrates a cross-section of anumber of different removal profiles that the tool 8 can have, dependingon the various controllable characteristics of the tool 8. As shown inFIG. 2d , the removal profiles illustrate the amount of material removedper unit dwell time. The actual removal within the contact area is givenby multiplying the removal profile by the time that the tool 8 works thesurface with that profile (the dwell time). By varying the dwell timesand the removal profile of the tool 8 (i.e. the controllablecharacteristics of the tool 8) as the tool 8 is passed over theworkpiece 5, the desired polishing is achieved.

A description will now be given of the way in which the computer controlsystem 9 determines how to control the various drive actuators of thepolishing machine to achieve the desired polishing effect along thepseudo-random tool-path. A description will then be given of variousways in which pseudo-random tool-paths can be defined.

Computer Control System

As discussed above, in this embodiment, the tool-path (i.e. the pathalong which the tool 8 moves relative to the workpiece 5) is fixed. Themain purpose of the computer control system 9 is therefore to select theoptimum control parameters to use to control the tool 8 at differentpositions along the defined tool-path, in order to remove the requiredamount of material from the workpiece 5 to obtain a workpiece having thedesired surface profile. FIG. 3 is a block diagram illustrating the maincomponents of the computer control system 9 used in this embodiment tocontrol the tool 8. As shown, the computer control system 9 includes atool-path effect processor 301 which receives:

-   -   (1) surface model data 303 defining the surface profile of the        workpiece 5 to be worked;    -   (2) tool-path data 305 which defines the path over the workpiece        5 along which the tool 8 will pass; and    -   (3) influence function data 307 which defines various influence        functions (representing the above-described removal profiles)        that the tool 8 can have on the workpiece 5, depending on the        values of the controlled variables.

As illustrated in FIG. 3, the function of the tool-path effect processor301 is to process this received data and to generate therefrom aninfluence matrix 309 which defines all the different possible influencesthat the tool 8 can have over the worksurface 5 depending on the valuesof the controlled parameters, for the given tool-path and workpieceprofile.

The computer control system 9 also includes an optimiser 311 which isdesigned to determine an optimised actuator parameter matrix 313 whichdefines the optimum control parameters to be used to control thepolishing apparatus as the tool 8 passes along the defined tool-path. Inthis embodiment, the optimiser 311 calculates the optimised actuatorparameter matrix 311 by performing an iterative optimisation routinebased on:

-   -   i) target profile data 315 which defines the target or desired        profile for the workpiece 5;    -   ii) a measured profile of the current workpiece 5 obtained from        a profile measurement sensor 317; and    -   iii) the influence matrix 309 calculated by the tool-path effect        processor 301.

As shown in FIG. 3, the optimised actuator parameter matrix 313calculated by the optimiser 311 is passed to a tool controller 315 whichalso receives the tool-path data 305. The tool controller 315 uses thisdata to generate the necessary control signals for application to thetool actuators 317 used to drive the components of the polishing machineto cause the tool 8 to remove the required material from the workpiece 5to generate a workpiece 5 having the desired (or target) profile. Asshown, the tool controller 315 also generates the control signals basedon sensor signals obtained from tool sensors 319 used to sense variousparameters (e.g. pressures, positions etc.) of the polishing machine.

Tool-Path Effect Processor

A more detailed description will now be given of the way in which thetool-path effect processor 301 calculates the above-described influencematrix 309. As discussed above, the tool-path effect processor 301receives the surface model data 303 defining the surface profile of theworkpiece 5 and tool-path data 305 which defines the path along whichthe tool 8 will pass relative to the surface of the workpiece 5. FIG. 4aschematically illustrates the surface of the workpiece 5 as defined bythe surface model data 303 and the tool-path 401 defined by thetool-path data 305. As shown in FIG. 4a , the tool-path 401 follows anon-periodic (in this case pseudo-random) pattern over the entiresurface of the workpiece 5. In this embodiment, the tool-path data 305includes a sequence of (x,y,z) coordinates (tool-path points) along thetool-path 401. Some of these tool-path points (TP_(i)) are shown in FIG.4a at the left-hand end of the tool-path 401.

In this embodiment, the surface model data 303 defines thethree-dimensional surface of the workpiece 5. The surface model data 303also includes the (x,y,z) coordinates of a number of mesh points(MP_(j)) which divide the surface into a number of triangular areas.Part of the mesh for the surface profile shown in FIG. 4a is shown inFIG. 4b . In this embodiment, the mesh is a by-product of the modellingtechnique used to model the surface of the workpiece 5—Delaunaytriangulation, which is a well-known technique used in computermodelling. In this embodiment, the surface model data 303 also includesdata defining an area that is associated with each of the mesh points.These areas are also a by-product of the modelling technique used tomodel the surface of the workpiece and are defined by the Voronoidiagram associated with the Delaunay triangulation. FIG. 4cdiagrammatically illustrates the area A_(MP2) that is associated withmesh point MP₂. A similar area will be defined within the surface modeldata 303 for each of the mesh points.

FIG. 4b also shows part of the tool-path 401 and illustrates (by thecircle 403) the contact area that the tool 8 will have on the surface ofthe workpiece 5 when it is brought into contact and centred over thefirst tool-path point (TP₁). As discussed above, the tool 8 will performpolishing of the surface of the workpiece 5 within this contact area403.

The purpose of the tool-path effect processor 301 is to consider whatpolishing effects (or influence) the tool 8 can have within the contactarea 403 as the area 403 moves when the tool 8 is moved along thetool-path 401. Clearly, it is not practical to consider the influence ofthe tool 8 on the surface 5 at all points within the circle 403 and forall positions along the tool-path 401. Therefore, the tool-path effectprocessor 301 only considers the influence that the tool 8 can have onthe worksurface at the mesh points (MP_(j)) which lie within the contactarea 403 when the contact area 403 is centred on each of the tool-pathpoints (TP_(i)) along the tool-path 401. To do this, the tool-patheffect processor 301 considers the various different removal profilesthat the tool 8 can have, which profiles are represented by a number ofinfluence functions defined by the influence function data 307. As thoseskilled in the art will appreciate, there are an infinite number ofdifferent removal profiles that the tool 8 can have, depending on thecontrollable characteristics of the tool 8 and on the surface profile ofthe workpiece 5. The influence functions defined by the influencefunction data 307 is a representative sample of the possible removalprofiles. The result of this processing by the tool-path effectprocessor 301 is the above-described influence matrix 309 which defines,for each tool-path point (TP_(i)) and each influence function profile,the influence (material removal per unit dwell time) that the tool 8will have at the mesh points in the local vicinity of that tool-pathpoint.

FIG. 5 is a flowchart illustrating in more detail the processing stepscarried out by the tool-path effect processor 301. As shown, in step s1,the tool-path effect processor 301 initialises a loop pointer, i,associated with the tool-path points (TP) defined by the tool-path data305 and a loop pointer, k, associated with the different influencefunctions defined by the influence function data 307. After these looppointers have been initialised, the processing proceeds to step s3 wherethe tool-path effect processor 301 calculates the normal, curvature andtangent of the surface of the workpiece 5 at the current tool-path point(TP_(i)) using the surface model data 303. At step s3, the tool-patheffect processor 301 also uses the calculated normal, curvature andtangent to deform the boundary of the contact area 403 for the currentinfluence function (identified by the loop pointer k).

The processing then proceeds to step s5, where the tool-path effectprocessor 301 centres the thus-deformed contact area 403 on the currenttool-path point (TP_(i)) identified by the loop pointer i. Theprocessing then proceeds to step s7 where the tool-path effect processor301 identifies the mesh points which lie within the thus positionedcontact area 403. For the contact area 403 shown in FIG. 4b , the meshpoints that would be identified in step s7 are: MP₂, MP₃, MP₈₉ andMP₁₂₃.

The processing then proceeds to step s9 where the tool-path effectprocessor 301 determines, for each of the mesh points identified in steps7, the amount of material that will be removed per unit dwell time. Theway in which the tool-path effect processor 301 makes this determinationwill be explained with reference to FIG. 6. In particular, FIG. 6 showsthe k^(th) influence function defined in the influence function data307. When determining the removal per unit dwell time for a mesh point,the tool-path effect processor 301 initially determines the distance (d)and direction of the current mesh point (MP_(j)) from the currenttool-path point (TP_(i)). The tool-path effect processor 301 then usesthis distance and direction to identify the part of the currentinfluence function which corresponds to the area (A_(mpj)) associatedwith the current mesh point. The tool-path effect processor 301 thencalculates the volume under the identified part of the current influencefunction. FIG. 6 illustrates the volume 601 under the influence functionthat is calculated in this way. The tool-path effect processor 301 thendivides this volume 601 by the area (A_(mpj)) associated with the meshpoint (MP_(j)) to determine an average removal per unit dwell time (inthis embodiment per microsecond), h_(i,j,k), for the current influencefunction (k).

The values thus determined are then stored in the influence matrix 309.FIG. 7 illustrates the form of the influence matrix 309 generated in theabove manner. As shown, the influence matrix 309 is a three-dimensionalmatrix storing average removals per unit dwell time (h_(i,j,k)) at themesh points positioned around each of the tool-path points for all thedifferent influence functions defined by the influence function data307.

After step s9, the processing proceeds to step s11 where the tool-patheffect processor 301 increments the value of the pointer (i) associatedwith the tool-path points. In step s13, the tool-path effect processor301 determines whether or not all of the tool-path points have beenprocessed in the above manner, by comparing the value of the pointer (i)with the total number of tool-paths points defined in the tool-path data305 (as defined by the variable TP_(MAX)). If all the tool-path pointshave not been processed in the above way, then the processing returns tostep s3 where the same influence function is deformed and repositionedover the next tool-path point and the above process is repeated.

Once all of the tool-path points have been processed in the above wayfor the current influence function, the processing proceeds to step s15where the tool-path effect processor 301 resets the pointer (i)associated with the tool-path points and increments the pointer (k)associated with the influence functions. In step s17, the tool-patheffect processor 301 determines whether or not the above processing hasbeen carried out for all of the possible influence functions defined inthe influence function data 307. As shown, this is achieved by comparingthe value of the loop pointer k with a variable K_(MAX) which definesthe total number of influence functions defined in the influencefunction data 307. If there are further influence functions to beprocessed, then the processing returns to step s3 where the aboveprocessing is repeated for the next influence function. Otherwise, theprocessing proceeds to step s19 where the tool-path effect processor 301stores the generated influence matrix 309 for use by the optimiser 311.The processing then ends.

Optimiser

FIG. 8 is a block diagram illustrating the main components of theoptimiser 311 used in this embodiment. As shown, the optimiser 311includes an optimiser engine 801 which carries out an iterativeoptimisation routine to determine the optimum actuator parameter valuesto be used at each tool-path point along the tool-path 401.

Based on a current actuator parameter matrix 803, a matrix combiner 805combines the information from this parameter matrix with the influencematrix 309 calculated by the tool-path effect processor 301, todetermine a removal map 807 defining the amount of material that will beremoved at the mesh points (mp_(j)) as the tool 8 passes along thedefined tool-path 401. In effect, therefore, the current actuatorparameter matrix 803 identifies, for each tool-path point, the influencefunction (k) that will be used at that point and the dwell time for thatpoint.

FIG. 9 schematically illustrates the form of the current actuatorparameter matrix 803. As shown, the parameter matrix 803 includes a setof tool parameters associated with each tool-path point (TP). Theparameters include the dwell time, the spot size (which defines thenominal size of the contact area 403), the pressure applied to the tool8, the value of the above-described angle θ etc.

When combining the current actuator parameter matrix 803 with theinfluence matrix 309, the matrix combiner 805 uses the tool parametersfor a current tool-path point (TP_(i)) to identify the influencefunction (k) for those parameter values. As those skilled in the artwill appreciate, the actuator parameter values defined in the currentactuator parameter matrix 803 for a tool-path point may not exactlycorrespond to one of the influence functions defined by the influencefunction data 307 (which are only a representative sample of allpossible influence functions). In this case, the matrix combiner 805interpolates the appropriate figures in the influence matrix 309 todetermine the appropriate removals per unit dwell time. The matrixcombiner 805 then multiplies the average removals per unit dwell timestored in the influence matrix 309 (or derived therefrom) with the dwelltime for the current tool-path point defined by the current actuatorparameter matrix 803, to give an estimated value for the removal at eachof the mesh points when the tool 8 is centred on the current tool-pathpoint (TP_(i)).

For example, for the actuator parameter matrix 803 illustrated in FIG.9a and for the first tool-path point (TP₁) the matrix combiner 805 woulduse the spot size (3.5), pressure (1), value of θ(5°) etc. to identifythe corresponding influence function corresponding to those parameters.Assuming for illustration that these actuator parameter valuescorrespond to the k^(th) influence function, then the matrix combiner805 looks up the influence matrix 309 to identify the mesh points storedfor the current tool-path point for the k^(th) influence function. Inthis example, the matrix combiner 805 would identify mesh points MP₂,MP₃, MP₈₉ and MP₁₂₃. The matrix combiner 805 then calculates theexpected removals at these mesh points by multiplying the removal perunit dwell time values (h_(i,j,k)) stored in the influence matrix 309for these mesh points with the dwell time defined for the firsttool-path point (TP₁). In this case, the matrix combiner 805 wouldcalculate the following expected removals:

-   -   h_(1,2,k)×1.2 for mesh point MP₂    -   h_(1,3,k)×1.2 for mesh point MP₃    -   h_(1,89,k)×1.2 for mesh point MP₈₉    -   h_(1,123,k)×1.2 for mesh point MP₁₂₃

The matrix combiner 805 performs a similar combination of the data fromthese two matrices for all of the tool-path points (TP_(i)). The matrixcombiner 805 then adds together all of the estimated removals that areobtained in respect of the same mesh point, to give a final estimatedremoval value for each mesh point. In particular, as can be seen fromFIG. 7, the mesh point MP₈₉ appears in the influence matrix 309 undertool-path point TP₁ and under tool-path point TP₂. Therefore, when theinfluence matrix 309 is combined with the current actuator parametermatrix 803, a removal value will be calculated for mesh point MP₈₉associated with tool-path point TP₁ and another removal value will becalculated for the same mesh point that is associated with tool-pathpoint TP₂. These two removals are then added together (together with anyother removal values for mesh point MP₈₉) to give the total removal formesh point MP₈₉.

FIG. 10 schematically illustrates the form of the final removal map 807that is calculated by the matrix combiner 805 as a result of the abovecalculations. As shown, the removal map 807 comprises a removal depth(in this embodiment in micrometers) associated with each mesh point.This removal map 807 identifies an estimated amount of material thatwill be removed by the tool 8 at the mesh points, if the tool is made tofollow the tool-path 401 (defined by the tool-path data 305) whilstusing the actuator parameter values defined in the current actuatorparameter matrix 803.

As shown in FIG. 8, in this embodiment, the calculated removal map 807is subtracted from a target removal map 809 which defines the amount ofmaterial that is actually desired to be removed from the workpiece atthe mesh points, in order to form a workpiece having the desired (ortarget) surface profile. (As shown in FIG. 8, the target removal map 809is calculated by subtracting the measured surface profile 811 of theworkpiece 5 from the target surface profile 315.) The resulting errormap 815 is then fed back into the optimiser engine 801 which uses thevalues in the error map 815 to update the values in the current actuatorparameter matrix 803. The above processing is then repeated using theupdated current actuator parameter matrix 803 until the optimiser engine801 determines that the optimisation process has fulfilled apredetermined convergence criterion. At this stage, the optimisation iscomplete and the optimiser engine 801 outputs the optimised actuatorparameter matrix 313 for use by the tool controller 315 in controllingthe driving and polishing of the tool 8 over the workpiece 5.

SECOND EMBODIMENT

In the first embodiment described above, the surface model data 303modelled the surface of the workpiece 5 to be worked in athree-dimensional, XYZ, coordinate space. Similarly, the tool-path data305 (which represents the path over the workpiece 5 along which the tool8 is to pass) also represented the tool-path 401 in three-dimensional,XYZ, coordinate space. In this second embodiment, the surface model data303 represents the surface of the workpiece 5 to be worked usingtwo-dimensional (bivarate) parametric curves and in particular usingnon-uniform rational B-splines (NURBS). Additionally, in thisembodiment, the tool-path data 305 also represents the tool-path 401using such non-uniform rational B-splines. The advantage of using NURBSto represent the surface of the workpiece 5 and to represent thetool-path 401 is that it reduces the complexity of the surface modellingand the complexity in defining the tool-path 401 as they are defined ina two-dimensional, UV, coordinate space. The way in which NURBS andother similar parametric modelling techniques can be used to modelsurfaces and curves can be found in the second edition of the standardtextbook entitled “Computer Graphics Principles and Practice” by Foleyet al. and in particular in chapter 11 thereof.

As will be apparent to those skilled in the art of surface modellingusing NURBS, the surface of the workpiece 5 is represented by parametriccurves and a plurality of control points (and knots) whose positions inthe UV space define the contours of the surface. For most surfaces beingmodelled, the control points will be non-uniformly distributed withinthe UV space, with more control points being provided in the regionsassociated with more complex parts of the surface being modelled. Inthis embodiment, the positions of mesh points that are used in theabove-described processing of the first embodiment, are determined bytransforming the UV coordinates of these control points (and knots) intocorresponding XYZ coordinates using the NURBS parametric polynomials.

FIG. 11 schematically illustrates the workpiece 1101 represented in theNURBS UV space and showing a non-uniform grid 1103 on which the controlpoints (CP_(j)) are located. FIG. 11 also illustrates the tool-path 1102when represented in the NURBS UV space. As will be apparent to thoseskilled in the art familiar with NURBS, the path 1102 will also berepresented by a two-dimensional array of control points (not shown),some of which may also be control points that define the surface profileof the workpiece 5. These control points for the path 1102 define aconnected sequence of spline segments which together form the 2D path1102. The tool-path points (TP_(i)) used in the above optimisationprocess are then determined from sample points (SP_(i)) taken alongthese spline segments. The positions of these sample points on thespline segments are chosen so that when the sample points (SP_(i)) aretransformed back into three-dimensional, XYZ, space using the NURBSpolynomials, the resulting tool-path points (TP_(i)) are approximatelyevenly spaced from each other along the tool-path 401.

As was described above, during the calculations performed by thetool-path effect processor 301, a volume under each influence functionwas calculated for each mesh point and this volume was then divided byan area associated with the mesh point. In the first embodiment, thearea was defined by the Voronoi diagram associated with the Delaunaytriangulation of the workpiece surface. In this embodiment, the areasassociated with the mesh points are defined by a second two-dimensionalgrid of grid points (defined in NURBS UV space) which is offset from thefirst grid 1103. This is illustrated in FIG. 12. In particular, FIG. 12shows some of the control points CP_(i) of the first grid 1103 as wellas the second grid 1201 which defines the areas 1203 surrounding thecontrol points of the first grid 1103. This is illustrated in FIG. 12 bythe shaded area 1203 in which control point CP₁ is located. As thoseskilled in the art will appreciate, the positions of the grid points (inUV space) which define the second grid 1201 and which define the areas1203 around the control points (CP_(i)) also have to be transformed intoXYZ space in order that the correct area in XYZ space can be calculatedand used in the above tool effect calculations.

Tool-Path Generation

There are various ways in which the tool-path data 305 can be generatedfor use in the above embodiments. The main criterion is that thetool-path 401 should follow the surface of the workpiece 5. This ispreferably done by calculating the positions of the above-describedtool-path points and by defining a straight line or a curved pathbetween successive tool-path points or by finding a best fit curve tothose points.

FIG. 13 is a block diagram illustrating a tool-path generator 1301 whichforms part of the polishing machine shown in FIG. 1 and which is used togenerate the above-described tool-path data 305 using the surface modeldata 303, rules stored in a rules database 1303 and a random numbergenerator 1305. In operation, the tool-path generator 1301 identifies aninitial tool-path point on the surface of the workpiece 5 to be worked.The tool-path generator 1301 uses rules within the rules database 1303to identify an appropriate starting point. For example, the rulesdatabase 1303 may define that the initial tool-path point should belocated in a certain position or area of the workpiece 5. The tool-pathgenerator 1301 then identifies the position of the next tool-path point,again referring to rules in the rules database 1303. For example, therules database 1303 may define that the tool-path generator shouldidentify the position of a subsequent tool-path point by randomlychoosing a direction to travel from a current tool-path point. In thiscase, the tool-path generator 1301 uses the random number generator 1305to make this random determination of the direction to be travelled.

In order to avoid a tool-path 401 which requires many different changesin direction of the tool 8, the rules database 1303 also preferablycontains rules limiting the frequency with which the tool-path generator1301 should change direction. As those skilled in the art willappreciate, this random selection of the direction to travel may resultin the tool-path generator 1301 identifying positions that do not lie onthe surface of the workpiece 5. In this case, the identified point in 3Dspace can be projected onto the surface in order to identify acorresponding tool-path point on the surface. The tool-path generator1301 continues in this manner either for a predetermined number of stepsor until it has defined tool-path data 305 which sufficiently covers thesurface of the worksurface 5.

As those skilled in the art will appreciate, various different types ofrules may be stored in the rules database 1303 in order to control thegeneration of the tool-path data 305 by the tool-path generator 1301.For example, in addition to the rules discussed above, the rulesdatabase 1303 may include the following rules:

-   -   1) rules which define what to do when the tool-path approaches        or encounters a boundary of the workpiece 5. For example, these        rules may define that the tool-path should approach the boundary        tangentially or at some predefined angle or range of angles or a        probabilistically defined range of angles, in order to control        the boundary conditions (edge effects). The tool-path may be        constrained so that, from a tool-path point which is within a        first predetermined distance of the workpiece boundary, the        tool-path may not proceed in a direction within a predetermined        angle range from the normal joining the workpiece boundary to        that tool-path point. Likewise, the tool-path may be constrained        so that, from a tool-path point on the workpiece boundary, the        tool-path may proceed in a direction along the boundary, or in a        direction away from the boundary and within a predetermined        angle range from the normal to the workpiece boundary.    -   2) Rules which control the tool-path generator 1301 in order to        limit the number of visits to each region of the workpiece 5.        These rules may be dynamically adjusted depending on the        workpiece 5 to be worked so that the number of visits allowed        within each region of the workpiece can vary depending on the        amount of material to be removed from those regions. In this        way, regions of the workpiece 5 which require less polishing are        allowed fewer visits whereas regions requiring more polishing        are allowed more visits. As those skilled in the art will        appreciate, to implement this rule, the tool-path generator 1301        will have to divide the worksurface into different regions and        will have to keep track of when the tool-path enters and exits        each region.    -   3) In addition to (2) above, the rules may also include rules to        control the direction in which the tool-path takes so that the        probability of visiting each region of the surface is a defined        function of the amount of material to be removed in that region.    -   4) Rules which define the minimum turning radius for the        tool-path (which is related to the maximum acceleration of the        machine tool).    -   5) Rules which define that the minimum bend radius over a        defined region of the surface is a deterministic or        probabilistic function of the distance from a specified point in        the surface (e.g. the centre). This rule ensures that the        tool-path 401 follows a relatively smooth path over the        worksurface.    -   6) Rules which define the number of times that the tool-path 401        can cross itself.

Various other rules will be apparent to those skilled in the art.

Instead of defining the tool-path 401 by identifying tool-path pointswithin a three-dimensional space, the three-dimensional surface modeldata 303 may be projected down onto a two-dimensional plane (e.g. theX,Y plane) and then the tool-path defined on that plane and, oncedefined, projected back into three-dimensional space. This significantlysimplifies the task of defining the tool-path as it is easier to definea tool-path in a two-dimensional plane rather than a three-dimensionalspace. However, with such a technique, the tool-paths becomeincreasingly distorted as the surface slope increases and fails when thesurface slope is perpendicular to the projection plane.

As described in the second embodiment, the tool-path may also be definedin the NURBS (or some other parametric) two-dimensional (UV) space. Thisis advantageous as it also reduces the problem to one of identifying apath in a two-dimensional plane and also avoids the problems associatedwith the above-described projections. In particular, when using NURBS todefine the surface of the workpiece 5, points in the UV plane which liewithin the boundary of the NURBS representation of the workpiece 5 willalways lie on the actual surface of the object in XYZ coordinate space.The method also preferably uses an “artificial ant” algorithm to definethe path in the UV plane.

The artificial ant algorithm may be used to define the positions ofactual path sample points (SP) within the UV NURBS space, which are thenconnected together by a straight line or a curve or which are used in acurve fitting algorithm to find a curve that best fits the identifiedpoints. Alternatively, the artificial ant algorithm can be used todefine the locations of control points which define the 2D path byspline segments such as NURBS. To carry out the artificial antalgorithm, the tool-path generator 1301 forms a grid or lattice ofpoints over the representation of the workpiece 5 in the UV plane. Thetool-path generator 1301 then starts at an initial lattice point andthen randomly choose to move left, right, up or down on the lattice.This process continues until the tool-path generator 1301 has definedsufficient points which cover the surface of the workpiece 5.

In the case where the chosen lattice points represent sample pointsalong the path (in UV space), the tool-path generator 1301 then convertsthese sample points into XYZ space to identify the XYZ tool-path points(TP_(i)). As those skilled in the art will appreciate, with thistechnique, in order to produce a relatively smooth tool-path thegridlines which form the lattice on which the artificial ant algorithmworks will have to be closely spaced. This technique is thereforerelatively time-consuming.

In the alternative case where the artificial ant algorithm defines thecontrol points for splines, the gridlines do not have to be as closelyspaced, although a non-uniform lattice may be preferable in order toensure that sufficient control points are provided in parts of theworkpiece which are geometrically complex or which require morepolishing. Further, as those skilled in the art will appreciate, if theartificial ant algorithm is used to define control points in order todefine the tool-path using splines, then the final sequence of splinesegments that is calculated has to be sampled along its length and thenthose sample points converted into XYZ space to identify the XYZtool-path points (TP_(i)).

It is not essential to define the tool-path 401 using some random walkor artificial ant algorithm in UV space. Instead, the path in UV spacemay be defined by providing a random distribution of control points usedto define a sequence of spline segments over the portion of the UV planecorresponding to the surface of the workpiece 5. Alternatively, insteadof being randomly distributed, the control points may be distributed ina more intelligent manner so that more control points are provided atany boundaries of the surface or at portions of the surface whichrequire more polishing.

As a further alternative, adaptive isocurves may be used to define thepath in the UV space. In particular, adaptive isocurves take advantageof the NURBS used to represent the surface to be polished. As discussedabove, every point on the surface of the workpiece 5 is represented bytwo coordinates: u and v. Using the NURBS polynomials, a (u,v)coordinate can be converted into a corresponding three-dimensionalcoordinate (x,y,z). By keeping either u or v constant, a functionplotted in UV space is a straight line. When this same function isexpressed in three-dimensional coordinates, it becomes a curve along thesurface. Therefore, calculations which would be very complicated inthree-dimensional space become simple actions involving only straightlines in UV space. These functions are called isocurves.

The simplest way to make an isocurve would be to choose one coordinate(U or V) to be constant and to let the other coordinate run from itsminimum to maximum value. The problem with using this method of definingisocurves becomes apparent when a hemisphere is considered. If u ischosen to be the coordinate which represents the direction of latitudeon the hemisphere, then v is the direction of longitude. If two sets ofisocurves are defined: one equally spaced in u and one equally spaced inv, then when these isocurves are translated into three dimensions ontothe surface of the hemisphere, the lines of longitude will still beequally spaced at each point, however, the lines of latitude will becloser together near the pole than at the equator of the hemisphere. Tosolve this problem, a more sophisticated version of isocurves can beused, called adaptive isocurves. Instead of running from one edge of thesurface to the other, adaptive isocurves start and stop so that everypoint on the surface is within a certain distance of one of thesecurves. In this case, every line of longitude on the hemisphere does notextend all the way to the pole.

These isocurves are produced recursively in the following manner.Initially, the algorithm starts with two isocurves defined in UV spaceat the opposite edges of the surface along lines where u or v isconstant. The desired spacing between the curves in XYZ space is thendefined, such as the radius of the polishing tool 8. The system thencalculates the distance between these isocurves (in XYZ space) at everypoint along the curves. If this distance is ever greater than thespecified spacing, a new isocurve (in UV space) is started at thatpoint, halfway between the two starting isocurves. When the distance isless than or equal to the defined spacing, the new isocurve is stopped.The algorithm then starts over, adding new isocurves between all theexisting isocurves until every point on the surface of the workpiece iswithin the defined spacing of at least one isocurve. The algorithm isthen repeated keeping the other of the u or v coordinates constant,resulting in two sets of perpendicular isocurves.

As those skilled in the art will appreciate, by defining the adaptiveisocurves using the tool radius for the spacing, every point on thesurface of the workpiece 5 will be within reach of the tool 8 if itfollows these isocurves.

An alternative method for creating a tool path that visits the entiresurface of a part but has no periodic pattern is to create a mazepattern. The tool can be considered to be wandering through a maze thattakes it over every point on the surface to be treated.

There are many types of mazes, and the type of maze that is the mostuseful in the present application is the unicursal maze. This type ofmaze has no dead ends, no crossings and no junctions, but is a singleconvoluted path leading from a start to a finish. Such mazes can beapplied in polishing, to cause a tool to “wander” over the entiresurface in a random path without ever crossing its past track. Aunicursal maze pattern can be generated very quickly and scaled up ordown to any size. In one embodiment, the first step in creating aunicursal maze tool path is to generate a maze. A second algorithm thensolves that maze and that solution is used as the tool path.

A perfect maze can contain junctions, loops and dead ends. Its definingcharacteristic is that any point in the maze can be reached from anyother point. To create a unicursal maze, we begin with a starting maze,which is a perfect maze having no loops.

The starting maze is converted into a unicursal maze by longitudinallydividing every passage, so that each passage becomes two adjacentpassages, and every dead end becomes a u-turn.

A unicursal maze may be generated by first dividing the region to betreated into a matrix of square cells, or by first covering the regionin hexagonal grid points. The effect of these two different startinggeometries affects the appearance of the tool path patterns that result.The “packing density”, which is the transverse spacing between adjacentstretches of the maze (and of the final tool path generated), depends onand can be set by, the size of the cells in the matrix, or the spacingbetween the hexagonal grid points. By varying the size of cells in thematrix, or varying the spacing between grid points, in predeterminedregions of the area to be treated, the “packing density” of the toolpath can be varied from one region to another of the area to be treated.In a particular embodiment, the size of cells in the matrix, or thespacing between grid points, may be reduced in regions adjacent the edgeof the area to be treated, so that the “packing density” of the toolpath is increased and adjacent stretches of the tool path are closertogether in these regions.

Square Unicursal Maze Tool Path

This method for generating a tool path uses an algorithm that dividesthe region to be treated into a matrix of square cells. The size of thecells is set on the basis of the polishing influence function of thetool. The matrix may be defined so that the pattern that results in theend will have the same average spacing between neighbouring pathsections as the raster or the spiral of a conventional polishing toolpath.

After subdividing the surface, the next step is the generation of thestarting maze. This may be done using the Aldous-Broder algorithm.Preferably, it starts in a corner cell and moves to a neighbouring cellat random, creating a step of the tool path. From there, it choosesanother neighbouring cell and moves to that cell if it has not beenpreviously visited, creating a second step. If every neighbour of a cellhas been visited, the algorithm randomly jumps to another visited celland again starts looking for unvisited neighbours. This creates ajunction in the starting maze. When every cell of the surface has beenvisited at least once, the maze is complete.

The Aldous-Broder algorithm is not the fastest method for producing thestarting maze, but it does have advantages, in that it can generate allpossible mazes of a given size and does so with equal probability ofproducing any particular maze pattern. The maze produced is random, withno bias towards any direction.

The starting maze, which comprises passage cells and wall cells, is thenconverted into a unicursal maze in three operations. In the firstoperation, all passage cells are converted into a third type of cellcalled “inner-wall”. In the second step, any wall cell now bordering aninner-wall cell is converted to a passage cell, creating a unicursalmaze. In the third step inner-wall cells are re-labelled as wall cells.

This produces a unicursal maze pattern, but not a tool path. Amaze-solving algorithm is used to solve the maze and store the solutionas a list of coordinates. These coordinates are used as control pointsfor a spline curve, and the spline curve is used as a tool path and isexpressed in NURBS space with coordinates ranging from 0 to 1 in u andv.

Hexagonal Unicursal Maze Algorithm

The first step in creating the unicursal maze tool path using thismethod is the generation of a first hexagonal grid. The spacing of thegrid is based on the spot size used for surface treatment. The spacingbetween adjacent stretches of the final tool path pattern is preferablyequal to half the maximum width of the polishing influence function.This first grid should have a spacing equal to twice the maximum widthof the polishing influence function, to make the final pattern densitywork out correctly.

A second hexagonal grid that is four times as dense as the first grid isthen overlaid on the first grid. This second grid must extend beyond thefirst grid by one row in each direction, to allow room to build thefinal pattern.

Next a minimal spanning tree is generated using the Aldous-Broder method(Aldous, D., SIAM Journal on Discrete Mathematics, 1990). This spanningtree is a starting maze which connects all of the points in the firsthexagonal grid.

To generate the spanning tree, a start point is chosen and allneighbouring points of the start point are identified. These neighbourscan be found, for example by considering all points that lie within adistance equal to the grid spacing as neighbours. An unvisited neighbouris randomly chosen and the algorithm moves to that point, creating thefirst step of the maze. Neighbours are again identified, and anunvisited neighbour is randomly chosen and the algorithm moves to thatpoint, creating the second step of the maze. As before, if allneighbours have been visited, the algorithm moves to a random visitedpoint elsewhere in the grid, creating a junction. This process isrepeated until every point in the large-scale first grid has beenvisited. For each step of the tree, the four fine-grid points of thesecond grid that the step passes through are identified. These pointsare saved in an array that holds the spanning tree pattern.

All points of the spanning tree pattern, and points of the fine gridpattern which are neighbours of points in the spanning tree pattern, arethen identified as a group (“collected”).

The points from this collected group which belong to the spanning treeare removed. The result is a unicursal maze pattern (Pullen, “ThinkLabyrinth” 2006), essentially a path which extends along one side of thespanning tree pattern.

As before, this pattern needs to be turned into a path made up of a listof coordinates using a maze-solving algorithm. A starting point isselected and marked as visited, then the neighbouring points in thepattern are identified, and the solver moves to the first unvisitedneighbour.

This process is repeated until every point in the pattern has beenvisited, which brings the solver back to the starting point. Thesepoints are then used as control points to create a spline curve. Thisspline curve is output as the tool path.

The hexagonal unicursal maze tool path can be generated for any shape,including parts with interior holes, by setting out the hexagonal gridsto correspond to the surface to be treated.

Tool-Path Selection

As those skilled in the art will appreciate, there are almost aninfinite number of possible tool-paths 401 which can be generated fortraversing the surface of the workpiece 5. Some of these tool-paths maynot be efficient or may leave artefacts on the surface of the workpiece5. However, hidden somewhere within all possible tool-paths, there willbe one or more tool-paths which combine the ideal qualities of polishingresults and efficiency. If one of these efficient tool-paths can beidentified, then it will simplify the optimisation process discussedabove and, in some cases, may remove the need for it altogether. Anumber of techniques which can be used to try to identify theseefficient tool-paths will now be described.

Genetic Algorithm

The tool-paths can be considered to be “states” of a system, and thecollection of all possible tool-paths is called the “state space”. Thereare many artificial intelligence techniques devoted to searching suchstate spaces for optimal state solutions. One of these methods is thegenetic algorithm, which can be described as searching the state spaceby randomly wandering through it. The inefficiency of this randomwandering is greatly decreased by creating an entire population ofcandidate solutions which are considered in parallel. The algorithmnarrows in on the best areas of the state space (i.e. the bestsolutions) by eliminating and replacing low quality candidate solutions.

Genetic algorithms model natural evolution by using a “survival of thefittest” philosophy. The algorithm begins with a population of candidatesolutions. Each of these solutions is evaluated and the weakestsolutions are eliminated. Characteristics of the best solutions arecombined to replace the eliminated candidates. The process is repeated anumber of times (or generations) until a solution is found whichsatisfies a predetermined criterion.

In the case of tool-path generation, the tool-path generator 1301initially creates the population of candidate tool-paths (in thisembodiment 100 candidate tool-paths). This is done firstly by dividingthe surface of the workpiece 5 (in the UV plane) into pixels (using theabove-described lattice) and then each candidate tool-path is created.The toolpath may be created by the unicursal maze methods, or byperforming a random walk (controlled using the random number generator1305) of 10,000 steps across these pixels on the workpiece surface. Thesystem then calculates a “dwell time map” for the workpiece 5 whichidentifies, for each of the above pixels, the amount of time that thetool 8 should spend at the point in XYZ space corresponding to thatpixel. The dwell time for each pixel depends on the amount of materialto be removed and is calculated by considering the difference betweenthe measured profile of the workpiece and the target profile of theworkpiece. When calculating the dwell time for each pixel, the tool-pathgenerator 1301 may assume that the polishing tool 8 has a constantaverage removal per unit dwell time.

The tool-path generator 1301 then calculates the velocity of the toolthrough each of the 10,000 steps for each candidate tool-path byaveraging the dwell times at the starting and ending pixels and bydividing the length of the step (in XYZ space) by this time.

After creating the population, the tool-path generator 1301 evaluateseach candidate tool-path. This evaluation is carried out, in thisembodiment, using the above-described tool-path effect processor 301 andthe optimiser 311. In particular, the tool-path generator 1301 passesthe candidate tool-paths (once transformed into XYZ space) to theabove-described tool-path effect processor 301 and the optimiser 311 sothat the above-described optimisation process is run for each candidatetool-path. At the end of the optimisation, the optimiser 311 passes thefinal error map 815 for each candidate tool-path back to the tool-pathgenerator 1301. The tool-path generator 1301 then compares these errormaps to identify the tool-paths which produce the lowest errors. Thetool-path generator 1301 also analyses the required velocity andacceleration of the tool 8 along each candidate tool-path. Based on theerror maps and the velocity and acceleration required for eachtool-path, the tool-path generator 1301 ranks the candidate tool-pathsand eliminates the bottom half of the candidate tool-paths.

The tool-path generator 1301 then randomly pairs the remaining 50candidate tool-paths. The tool-path generator 1301 then arbitrarilyselects a step in each tool-path of each pair and then both tool-pathsare “cut” at that point. This is schematically illustrated in FIG. 14awhich shows a pair of candidate tool-paths 401-1 and 401-2 andillustrating the point 1401-1 and 1401-2 at which the two tool-paths arecut. The tool-path generator 1301 then splices together the left-handportion of the first tool-path 401-1 with the right-hand portion of thesecond tool-path 401-2 and splices together the left-hand portion of thesecond tool-path 401-2 with the right-hand portion of the firsttool-path 401-1, to thereby generate two new candidate tool-paths 401-3and 401-4 as shown in FIG. 14b . This cutting and splicing is performedfor each pair of candidate tool-paths. This cutting and splicing willtherefore generate another 50 possible candidate tool-paths which arecombined with the top 50 candidate tool-paths that were ranked, tocreate a new population of 100 candidate tool-paths. This whole processis then repeated a number of times (in this case 50). The tool-pathgenerator 1301 then uses the best candidate tool-path to define thetool-path data 305 to be used to control the actual polishing of theworkpiece 5.

Stipple Tool-Path Algorithm

A number of techniques have been described above for generating apseudo-random tool-path which passes over substantially the whole of theworksurface 5. However, a pseudo-random tool-path which crosses itselfintroduces the further constraint that no matter how many times a pathcrosses the same point on the worksurface, the total amount of materialremoved at that point must equal the desired amount of material (asdefined by the above-described error map). Therefore, if the tool-pathcrosses a point on the surface more than once, either the time that thetool spends at that point or the influence function (or both) must becontrolled each time the tool passes over that point, so that thecumulative effect of the tool does not exceed the required treatment atthat point. This may create an obstruction in producing a smoothtool-path where the velocity of the tool 8 is smoothly varied over theworkpiece 5.

Additionally, the random tool-path must not contain instructions whichwould be impossible for the polishing machine to follow. In other words,the tool-path cannot include extreme changes in velocity and direction.For most workpieces 5, the difference in the amount of time that has tobe spent in adjacent parts of the surface is not great. Therefore, ifthe dwell time at a crossing point is divided equally between the twoparts of the tool-path, then as the polishing machine follows the path,it will suddenly be asked to pass through that crossing point veryquickly compared to the speed at neighbouring pixels. Not only might theacceleration required be more than the polishing machine can carry out,this quick change in velocity will also result in a polishing artefacton the surface of the workpiece 5. Therefore, if the random path isallowed to wander across the surface of the workpiece long enough toproduce reasonable removal results, the integrated effect of thesplitting of the dwell times at the crossing points, results in astructure pattern which outlines the edges of the pixels on theworkpiece defined by the dwell time map.

One way to deal with this problem is to avoid it entirely. The inventorshave realised that they can do this by using techniques developed forthe quilting industry. In particular, when a quilt is made, the toppiece of the quilt is sewn to a backing piece of cloth with a layer ofcotton batting sandwiched between them. A pattern is then stitched overthe surface of the entire quilt to prevent the batting from bunchingtogether and becoming lumpy. Quilters have developed a pattern forefficiently covering the entire quilt surface with a roughlyequally-spaced amount of stitching called “stippling”. The stippling isdesigned to cover the entire quilt using a continuous smooth path ofstitches which never crosses itself. The inventors have realised thatthese stippling patterns have many characteristics which are desirablefor machine tool purposes. In particular, they are pseudo-random, theynever cross themselves, they are continuous and they define a smoothpath. FIG. 15 schematically illustrates an example of a stipplingpattern 1501.

As these stippling patterns are designed to be applied to asubstantially planar object (the quilt), the same stippling pattern canbe used to fill the area defined by the workpiece 5, for example, in theabove-described NURBS UV space. Once defined in this way, the actual 3Dtool-path can be identified by using the NURBS polynomials to transformthe thus defined tool-path back into XYZ space.

The inventors have also realised that one problem with using the abovestippling patterns for some workpieces 5 is that the density of thestippling pattern may not be enough to accomplish the required amount ofpolishing without making the stippling pattern over convoluted,resulting again in the problems of exceeding the polishing machine'sacceleration limits. This problem can, however, be overcome by splittingthe polishing process into several phases. FIG. 16 is a block diagramillustrating the main components of a tool-path generator 1601 whichoperates in this manner.

As shown, the tool-path generator 1601 includes a stippling patterndatabase 1603 which includes data defining a plurality of differentstippling patterns (each similar to the one shown in FIG. 15). Thetool-path generator 1601 also includes a removal map calculator 1605which selects (preferably randomly) one of the stippling patterns fromthe stippling pattern database 1603 and uses it and the dwell time map1607 for the workpiece 5 to be worked, to generate a predicted removalmap 1609 which represents the predicted material removal by passing thetool 8 over the workpiece 5 using a tool-path corresponding to theselected stippling pattern. This predicted removal map 1509 is thensubtracted from the target removal map 809 (obtained by subtracting thetarget surface profile for the workpiece 5 from the measured surfaceprofile) to generate an error map 1511 identifying the residual materialstill to be removed. The removal map calculator 1505 then uses thiserror map to calculate a new dwell time map defining the time to bespent at each pixel on the surface in order to remove this residualmaterial.

The removal map calculator 1605 then selects another stippling patternfrom the stippling pattern database 1603 and uses it and the new dwelltime map that has been calculated to generate a new predicted removalmap 1609. This new predicted removal map 1609 is then subtracted fromthe target removal map 809 (which has been updated so that it is thesame as the error map 1611 obtained from the first removal). Thisgenerates a new error map 1611 which is passed to the removal mapcalculator 1605 as before. Once the removal map calculator 1605determines that the error map is sufficiently close to zero, the removalmap calculator 1605 passes the tool-path data for the selected stipplingpatterns together with their associated dwell time maps to the toolcontroller 315 which uses this data to control the polishing machine toperform the actual polishing of the workpiece 5.

As those skilled in the art will appreciate, the removal map calculator1605 may be arranged to simulate the effect of different combinations ofstippling patterns to identify the most efficient combination ofstippling patterns to be used for a given workpiece 5. In this way, thedesired polishing can be achieved using the minimum number of stipplingpatterns and/or with a desired quality of polishing finish.

As a further alternative, when the removal map calculator 1605 selects astippling pattern from the stippling pattern database 1503, it maymodify the stippling pattern in order to make the tool-path definedthereby more efficient for the particular workpiece 5 to be worked. Inparticular, the dwell time map 1607 identifies the points on the surfaceof the workpiece at which the tool 8 should spend the most time. Theremoval map calculator 1505 can therefore use this information from thedwell time map to vary the density of the stippling pattern over theworkpiece 5 so that, the greater the density, the more time is spent bythe tool.

For example, if each stippling pattern 1501 is defined by aself-avoiding random walk defined on a lattice of points distributedover the surface of the workpiece (as represented in UV space or aprojection), then the removal map calculator 1605 can vary the densityof the stippling pattern by varying the density of this lattice, so thatthe gridlines of the lattice are closer together in areas which requiremore polishing and further apart elsewhere. The chosen walk sequence canthen take place on this modified lattice. A smooth tool-path can then bedefined using the visited points on the lattice as control points forspline segments which represent the tool-path (in UV space) in themanner described above.

Modifications and Alternatives

Although all of the embodiments described above have related to theoperation of a polishing machine which removes material from the surfaceof a workpiece 5, the inventions described in the present applicationare also applicable to many other types of machine tools. For example,the inventions described above are applicable to machine tools which areused in any physical process that adds material to the surface of theworkpiece 5, removes material from the surface of the workpiece 5 ormodifies the surface (or a region under the surface) of the workpiece 5in some way or to a tool-path that is used in any physical process whichemploys any combination of these effects. Processes which add materialinclude those which apply a coating or a plating to the surface of theworkpiece. Other processes which remove material from the surface of theworkpiece include grinding, ion abrasion etc. and processes which modifythe surface of the workpiece include hardening, ion implantation,radioactive bombardment etc. The above systems can be used in anyprocess where there is a tool which is to be moved across the surface ofa workpiece along a tool-path and where the tool has a time-dependentcumulative effect within a local area on or under the surface of theworkpiece. In many applications, the tool will physically touch thesurface of the workpiece, whereas in others it may provide the action ata distance from the surface (for example, a tool which outputs a laserbeam onto the surface).

In the above embodiments, a tool-path was used which was non-periodic.The use of a non-periodic tool-path overcomes the problem associatedwith periodic tool-paths (such as rasters or spirals) in which periodicartefacts are left on the surface of the workpiece. The tool path may bedefined so that, as the tool passes along the tool path over the surfaceto be worked, the transverse distance between any section of thetool-path and a neighbouring section beside it is made to lie within apredetermined range of distances. This range may be dependant on the“footprint” of the tool, and may be set so that a predetermined degreeof overlap between the footprint of the tool as it passes along thefirst section of the tool-path and the footprint of the tool as itpasses along the neighbouring section of the tool-path. Preferably aminimum degree of overlap is maintained, at all positions along thetool-path, to ensure treatment of all parts of the workpiece surface. Inalternative embodiments, the tool path may be defined so that thetransverse distance between neighbouring sections of the tool path isreduced at points along the path, and in a particular embodiment inregions adjacent the edge of the workpiece (or the edge of the area ofthe workpiece to be treated). This reduction in separation betweenneighbouring sections of the tool path may be accompanied by a reductionin the footprint of the tool used at these regions of the tool path.

In the first and second embodiments described above, a tool-path wasdefined in advance and then used in an optimisation process to determinethe optimum control parameters for controlling the polishing machine. Asthose skilled in the art will appreciate, in an alternative embodiment,the tool-path may not be defined in advance and may also be optimised inthe optimisation process. The advantage of this technique is that theoptimisation process would determine the optimum tool-path as well asthe optimum control parameters for the polishing machine. Thedisadvantage, however, is that the optimisation process would takesignificantly longer due to the increased calculations that would berequired for the optimisation process to converge on an optimaltool-path. Further, in such an embodiment, the above-described influencematrix would have to be calculated at each step of the optimisationiteration as the optimiser changes the control parameters and thetool-path.

In the second embodiment described above, the tool-path was definedusing NURBS. As those skilled in the art will appreciate, other types ofsplines can be used to define the tool-path. As mentioned above, the useof splines is advantageous because the control points which define thesplines are defined in a two-dimensional coordinate space. Various othertypes of splines which can be used to represent the tool-path aredescribed in chapter 11 of the above-mentioned textbook “ComputerGraphics Principles and Practice” by Foley et al.

In the first embodiment, the tool-path was predefined and a tool-patheffect processor was used to calculate the influence matrix 309. Asthose skilled in the art will appreciate, one of the advantages ofpre-calculating the influence matrix in this way is that it speeds upthe optimisation process performed by the optimiser 311. However, it isnot essential to pre-calculate the influence matrix 309 in this way. Forexample, the optimiser 311 may be arranged to use the control parametersat each tool-path point to work out the exact influence that the tool 8will have at that point and at the surrounding mesh points.

In the first embodiment described above, the tool controller 315received sensor inputs from tool sensors 319. These sensors wereprovided to allow the tool controller 315 to control the actuators usingconventional feedback control strategies. However, as those skilled inthe art will appreciate, it is not essential for the tool controller 315to use the feedback from the tool sensors 319.

In the first embodiment described above, the tool 8 had a deformablesheet of polishing material 110 attached to its end. As discussed above,this sheet deformed when the tool was brought down into contact with thesurface to be polished. As a result of this deformation, the influencefunctions were also deformed based on the surface characteristics of theworkpiece to be worked within the contact area. As those skilled in theart will appreciate, if the tool 8 has a rigid polishing head, then nodeformation will occur and therefore, the tool-path effect processor 301does not need to consider the deformation when the tool is brought downinto contact with the surface.

In the first embodiment described above, the surface of the workpiece 5was modelled using Delaunay triangulation techniques. As those skilledin the art will appreciate, other surface modelling techniques can beused, details of which can be found in any standard text.

In the first embodiment described above, the polishing machine used apolishing head having a sheet of polishing material attached to its end.As those skilled in the art will appreciate, during the polishingoperation, water or other lubricant can be added to the surface in orderto cool the machine tool. Further, abrasive particles may also beprovided within the lubricant to help in the polishing process. For amore detailed description of the various techniques that can be used topolish a workpiece, the reader is referred to WO 00/32353.

In the first embodiment described above, during the calculation of theinfluence matrix, the tool-path effect processor 301 calculated a volumeunder each influence function based on an area associated with a meshpoint under consideration. As those skilled in the art will appreciate,in a simpler embodiment, the tool-path effect processor may simplydetermine the value of the influence function at the mesh point.However, calculating the volume under the influence function ispreferred as this generates a value representing the average removal inthe area surrounding the mesh point.

In the above embodiment, the influence matrix 309 generated by thetool-path effect processor was a three-dimensional table representingthe removal rates at each mesh point as the tool progresses along thetool-path. As those skilled in the art will appreciate, this informationcan be represented in a number of different data structures. It is notessential that it is represented in a three-dimensional matrix.

In the first embodiment, the optimiser subtracted the removal mapobtained from the matrix combiner from the target removal map to definean error map that was used to drive the optimiser engine 801. As thoseskilled in the art will appreciate, the samples in the target removalmap 809 may not correspond exactly to the positions of the mesh points.In this case, the optimiser would interpolate the samples within thetarget removal map to determine appropriate target removal values ateach of the mesh points. The optimiser can then simply subtract theestimated removal from the target removal for each of the mesh points togenerate the desired error map 815.

In the above embodiments, the workpiece to be worked by the machine toolwas supported on a support which could move the tool relative to thetool head. As those skilled in the art will appreciate, the workpiecemay be supported by a fixed support, in which case, the tool head willhave to be mounted on an appropriate arm which can move in the requireddirections to follow the tool-path. Alternatively still, the tool headmay be fixed and the support may be arranged to move the workpiece inall of the directions in order to allow the tool to follow thetool-path.

In the above embodiment, the influence functions were defined in termsof the amount of material removed per unit dwell time. As those skilledin the art will appreciate, instead of storing data in terms of suchdwell times, the influence function data can be stored in terms ofmaterial removal per unit of speed of the tool 8. Both representationsare equivalent as the speed is related to the dwell times by thedistance between the points to which the dwell times relate.

The above-described embodiments relate to methods in which the amount oftreatment to be delivered to the workpiece surface varies from place toplace over the surface of the workpiece. Treatment may be concentratedat a particular area of the workpiece either by slowing down the tool inits movement along the tool path when the tool path enters an arearequiring more concentrated treatment. The “packing density” of the toolpath may be increased at areas requiring more concentrated treatment, byreducing the spacing between neighbouring stretches of the tool path inthese areas. Concentrated treatment may also be delivered by increasingthe influence function of the tool as it passes along the tool path inthe area requiring concentrated treatment (for example by increasing thepressure of the tool against the workpiece to increase the rate ofremoval of material), while maintaining a constant tool speed along thepath. It is further foreseen that combinations of two or more of thesethree techniques for providing concentrated treatment may be usedtogether during the treatment process. Likewise, by speeding up thetool, increasing spacing between neighbouring stretches of the toolpath, and/or decreasing the influence function of the tool a lessconcentrated treatment may be delivered at points along the tool path.

In an alternative embodiment of the surface treatment method, a toolpath may be laid out on a surface to be treated, and the treatment toolmay then be moved along the tool path over the surface, keeping theinfluence function of the tool constant. This treatment method findsutility in surface treatment processes where a uniform treatment is tobe delivered to all parts of the surface, for example to change asurface characteristic without changing the overall shape or form of thesurface.

Where the tool path is of a type having no crossing points, this willresult in a uniform surface treatment to all sections of the surface,for example removing or applying a uniform layer of material from or tothe surface. The result will be a workpiece having the same form as theoriginal workpiece, but with a surface texture or other surfacecharacteristic which has been uniformly modified by the passage of thesurface treatment tool.

Where the tool path is of a type which has crossing points, such as maybe generated by a “random walk” algorithm, an effective treatment of thesurface may still be made provided that crossing points of the tool pathare distributed substantially evenly over the surface, so that nolocalised concentration of crossing points occurs. If crossing pointsare concentrated in a particular area of the surface, then excessivetreatment is delivered to that area, and for example excessive amountsof material may be removed or deposited such that the form of thesurface is in fact altered.

In treatment processes of this type, there is the requirement firstly tomeasure the surface to be treated, in order to generate a tool pathwhich follows the measured surface. However, there is no need togenerate coordinates of a target workpiece surface, nor an error map toquantify the differences between the actual surface of the workpiece andthe target surface, so processing time is markedly reduced.

In the embodiments described above, various processes and algorithms aredescribed. These processes and algorithms may be performed usingdedicated hardware circuits but are preferably performed using softwaremodules which control the operation of a programmable processor, such asa personal computer, workstation or the like. The software may beprovided on a recording medium such as a CD-ROM or it may be provided ona carrier signal downloaded over a computer network (such as theInternet). The software may be provided in any computer language and incompiled or in uncompiled form.

What is claimed is:
 1. An apparatus for performing a surface treatmenton an area of a workpiece, the apparatus comprising: a surface treatmenttool having a treatment footprint; a tool-path generator comprising: amemory configured to store surface model data defining athree-dimensional (3D) surface area of a workpiece to be worked; a rulesdatabase comprising a plurality of rules relating to tool-pathgeneration; and a tool-path determiner configured to determine datarepresenting a 3D tool-path along the 3D surface of the workpiece areato be worked in accordance with rules from said rules database, whereinsaid surface model data defines said surface in a two-dimensional (2D)projection and wherein said tool-path determiner is configured todetermine data representing a 2D path within said 2D projection and isconfigured to project the data representing said 2D path into 3D todefine said data representing said 3D tool-path; and means to move thesurface treatment tool relative to the workpiece along the determined 3Dtool path such that the footprint of the treatment tool passes over allpoints of the area.
 2. An apparatus according to claim 1, wherein saidsurface model data is configured to define said surface usingtwo-dimensional parametric polynomials, wherein said tool-pathdeterminer is configured to determine data representing a 2D path withina 2D space of said parametric polynomials and is configured to use saidparametric polynomials to convert said data representing said 2D pathinto said data representing said 3D tool-path.
 3. An apparatus accordingto claim 2, wherein said tool-path determiner is configured to determinesaid data representing said 2D path using one or more predeterminedstippling patterns.
 4. An apparatus according to claim 2, wherein saidtool-path determiner is configured to determine said data representativeof said 2D path using adaptive isocurves in accordance with rules insaid rules database.
 5. An apparatus according to claim 2, wherein saidtool-path determiner is configured to determine said data representativeof said 2D path using a random walk algorithm in accordance with rulesin said rules database.
 6. An apparatus according to claim 2, whereinsaid tool-path determiner is configured to determine said datarepresentative of said 2D path using an artificial ant algorithm inaccordance with rules in said rules database.
 7. An apparatus accordingto claim 2, wherein said tool-path determiner is configured to determinesaid data representative of said 2D path on the basis of a mazealgorithm in accordance with rules in said rules database.
 8. Anapparatus according to claim 2, wherein said tool-path determiner isconfigured to determine said data representative of said 2D path using agenetic algorithm in accordance with rules in said rules database.
 9. Anapparatus according to claim 2, wherein said data representative of said2D path comprises a sequence of 2D points and wherein the tool-pathdeterminer is configured to determine said data representing saidpseudo-random tool-path by converting said sequence of 2D points into 3Dpoints using said parametric polynomials.
 10. An apparatus according toclaim 9, wherein said tool-path determiner is configured to determinelines and/or curves between successive 3D points obtained from saidparametric polynomials to determine said data representing said 3Dtool-path.
 11. An apparatus according to claim 9, wherein said tool-pathdeterminer is configured to fit a curve to said 3D points in order todetermine said data representing said 3D tool-path.
 12. An apparatusaccording to claim 2, wherein said tool-path determiner is configured todetermine a sequence of points within said 2D space, which 2D pointsidentify control points for a plurality of spline segments that definesaid 2D path.
 13. An apparatus according to claim 12, wherein saidsurface model data represents said surface using two-dimensionalnon-uniform rational B-splines and wherein said control points arecontrol points for a non-uniform rational B-spline.
 14. An apparatusaccording to claim 2, wherein said data representative of said 2D pathcomprises 2D coordinate positions for a sequence of points randomlypositioned within said 2D space.
 15. An apparatus according to claim 2,wherein said tool-path determiner is configured to determine datarepresenting said 2D path comprising a sequence of points positioned insaid 2D space in accordance with a distribution function defined byrules in said rules database.
 16. An apparatus according to claim 15,wherein said rules define a weighted distribution function which causessaid tool-path determiner to determine more 2D points at positions insaid 2D space corresponding to boundaries of said area of saidworkpiece.
 17. An apparatus according to claim 2, wherein said surfacemodel data includes data identifying an amount of work to be carried outon different parts of said area of said workpiece and wherein saidtool-path determiner is configured to determine data representing said2D path comprising a sequence of points distributed within said 2D spacein dependence upon the amount of work to be carried out in the differentparts of the area.
 18. An apparatus according to claim 2, wherein saidtool-path determiner is configured to determine data representative of a2D path which does not cross itself within said 2D space.
 19. Anapparatus according to claim 2, wherein said tool-path determiner isconfigured to determine data representing said 2D path using auniformly-spaced lattice of points within said 2D space.
 20. Anapparatus according to claim 2, wherein said tool-path determiner isconfigured to define said data representative of said 2D path using alattice of points in which the spacing between lattice points depends onan amount of work to be performed in a corresponding portion of saidsurface.
 21. An apparatus according to claim 2, wherein said tool-pathdeterminer is configured to determine data representing a plurality of2D paths and wherein said tool-path definer is configured to select oneor more of said 2D paths in accordance with a predetermined selectionprocess.
 22. An apparatus according to claim 21, wherein said tool-pathdeterminer is configured to select said one or more 2D paths usingartificial intelligence techniques in accordance with rules in saidrules database.
 23. An apparatus according to claim 21, wherein saidtool-path determiner is configured to evaluate estimated effects of atool working said surface along tool-paths corresponding to eachpossible 2D path and configured to select one or more of said 2D pathson the basis of that evaluated estimated effects.
 24. An apparatusaccording to claim 2, wherein said tool-path determiner is configured todetermine data representing said 3D tool-path which comprises 3Dcoordinates of a sequence of points along said tool-path.
 25. Anapparatus according to claim 24, wherein said tool-path determiner isconfigured to determine lines and/or curves between successive pointswithin said sequence of points in order determine said data representingsaid 3D tool-path.
 26. An apparatus according to claim 24, wherein saidtool-path determiner is configured to fit a curve to said sequence ofpoints in order to determine said data representating said 3D tool-path.27. An apparatus according to claim 1, wherein said tool-path determineris configured to determine data representing a pseudo-random 3Dtool-path.
 28. An apparatus according to claim 1 wherein said 3Dtool-path does not cross itself.
 29. An apparatus according to claim 1,wherein said tool-path determiner is configured to determine datarepresenting a pseudo-random tool-path in which the transverse distancebetween neighbouring stretches of the path is arranged to be within apredetermined range of distances.
 30. An apparatus according to claim 1,wherein said tool-path determiner is configured to determine datarepresenting a 3D tool-path in which the transverse distance betweenneighbouring stretches of the path is substantially the same for all orsubstantially all points on the path.
 31. An apparatus according toclaim 1, wherein the surface treatment tool is capable of working on orunder the surface of a workpiece.
 32. An apparatus according to claim 1,wherein the 3-D tool-path is arranged so that, from a tool-path pointwhich is within a first predetermined distance of a boundary of thearea, the tool-path may not proceed in a direction within apredetermined angular range from the normal joining the boundary to thattool-path point.
 33. An apparatus according to claim 1, wherein thetool-path is arranged so that, from a tool-path point on a boundary ofthe area, the tool-path may only proceed in a direction away from theboundary and within a predetermined angular range from the normal to theworkpiece boundary.
 34. An apparatus according to claim 1 wherein saidmeans to move the surface treatment tool relative to the workpiececomprises a controller configured to control relative movement betweenthe workpiece and the tool to cause the tool to follow the 3-D tool-pathacross said surface of the workpiece to effect a time-dependentcumulative action on or under said surface.
 35. An apparatus accordingto claim 34, wherein said controller is configured to control the valuesof one or more controllable parameters of the apparatus to control thetreatment of the workpiece along said tool-path, to cause the apparatusto vary the treatment of said surface at different locations along saidtool-path.
 36. An apparatus according to claim 35, wherein thecontroller comprises an optimiser configured to optimise controlparameters used to control the working of the workpiece along saidtool-path.
 37. An apparatus according to claim 36, wherein the surfacetreatment tool is a tool for removing material from the workpiecesurface, and wherein said optimiser is configured to use data definingan influence function of the tool, and data defining a desired profileand a current profile of the surface of the workpiece.
 38. An apparatusaccording to claim 37, wherein said controller is configured todetermine dwell times or tool speeds for predetermined positions on saidsurface of the workpiece to provide a predicted removal profile and isconfigured to vary the dwell times or tool speeds and other controllableparameters of the tool using a numerical optimisation process todetermine control parameters and dwell times or tool speeds along saidtool path that will minimise an estimated error between said desiredprofile and the profile of the surface after said tool has performedsaid polishing or abrading along said tool path.
 39. An apparatusaccording to claim 1, wherein said treatment comprises one or more of:the removal of material from the surface of said workpiece; abrading orpolishing the workpiece; the addition of material to the surface of saidworkpiece; and the modification of the surface or a region under thesurface of said workpiece.
 40. A method for performing a surfacetreatment on an area of a workpiece by moving a surface treatment devicehaving a treatment footprint over the area; the method comprising:generating a 3D tool-path which visits all points of the area, in atool-path generator comprising: a memory configured to store surfacemodel data defining a three-dimensional (3D) surface of the area to beworked; a rules database comprising a plurality of rules relating totool-path generation; and a tool-path determiner configured to determinedata representing a 3D tool path along the surface of the area of theworkpiece to be worked in accordance with rules from said rulesdatabase, wherein said surface model data defines said surface in atwo-dimensional (2D) projection and wherein said tool-path determiner isconfigured to determine data representing a 2D path within said 2Dprojection and is configured to project the data representing said 2Dpath into 3D to define said data representing said 3D tool-path; andmoving the surface treatment device along the 3D tool-path such that thefootprint of the treatment tool passes over all points of the area. 41.A method according to claim 40, wherein the path is a random orpseudo-random path.
 42. A method according to claim 40, wherein the 3Dtool-path visits all points of the area without visiting any pointtwice.
 43. A method according to claim 40, wherein the transversedistance between neighbouring stretches of the 3D tool-path is arrangedto be within a predetermined range of distances.
 44. A method accordingto claim 40, wherein the 3D tool-path is arranged so that the transversedistance between neighbouring stretches of the path is substantially thesame for all or substantially all points on the path.
 45. A method ofmanufacturing a semiconductor device comprising shaping or treating asilicon wafer using a method according to claim 40, and manufacturingthe semiconductor device from the shaped or treated wafer.
 46. A methodof manufacturing a display device comprising shaping or treating ascreen component using a method according to claim 40, and including thescreen component in the display device.
 47. A method of manufacturing anoptical device comprising shaping or treating an optical component usinga method according to claim 40, and manufacturing the optical deviceusing the shaped or treated optical component.
 48. A method according toclaim 47, wherein the optical component is a lens or a mirror.